The logic behind combinations As an example, calculate the number of $5$ card hands possible from a standard $52$ card deck. 
Using the combinations formula, 
$$= \frac{n!}{r!(n-r)!}$$
$$= \frac{52!}{5!(52-5)!}$$
$$= \frac{52!}{5!47!}$$
$$= 2,598,960\text{ combinations}$$
I was wondering what the logic is behind combinations? Is it because there are 52 cards to choose from, except we're only selecting $5$ of them, to which the person holding them can rearrange however they please, hence we divide by $5!$ to account for the permutations of those 5 cards? Then do we divide by $47!$ because the remaining cards are irrelevant?
 A: Consider drawing $1$ card at a time.
The first card can be any of the $52$ cards.
The second can be any of the remaining $51$.
The third can be any of $50$... etc
So you have $52\times 51\times 50\times 49\times 48$ possibilities for $5$ cards. This is more conveniently written $\dfrac{52!}{47!}$.
But now you're counting the same hand $5!$ times because there are $5!$ ways of arranging $5$ cards. Dividing by $5!$ gives $\dfrac{52!}{5!47!}$
A: $n!$: the number of ways to select $n$ cards (from $n$ cards), such that order matters.
$\frac{n!}{(n-r)!}=n(n-1)(n-2)\cdots (n-r+1):$ the number of ways of selecting $r$ cards from $n$ cards such that order matters.
$r!$: The number of possible orderings of $r$ cards.
We don't want to account for order, hence we divide the second by the third as you said.
A: There are $52!$ distinct ways to shuffle a deck of $52$ cards.
However, we don't care about the order of the top $5$ or the bottom $47$ cards, we only wish to count ways to select distinct sets of cards for the hand.
Every deck is one of a group of $5!$ which differ only by the order of the top $5$ cards.   Every deck is also one of a group of $47!$ which differ only by the order of the bottom $47$.   Therefore every deck is one of a group of $5!42!$ which differ only by the order of the top $5$ and bottom $47$ cards.
So there are ${^{52!}\!{\big/}_{5!\,47!}}$ distinct ways to deal a hand of $5$ cards from a deck of $52$.
